<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN">
<HTML>
<HEAD>
<meta name="GENERATOR" content="Microsoft&reg; HTML Help Workshop 4.1">
<LINK REL="stylesheet" HREF="Orbiter.css" TYPE="TEXT/CSS">
<Title>Kepler's laws</Title>
</HEAD>
<BODY>
<h1>Kepler's laws of planetary motion</h1>
<p>
From the study of observational data collected by astronomer Tycho Brahe, Johannes Kepler was able to infer his famous three laws of planetary motion, which can be stated as follows:

<h2>Kepler's first law</h2>
<table cols=2 width=100% cellpadding=5><tr>
<td><b>Each planet moves in an ellipse with the Sun at one focus.</b></td>
<td><img src="kepler1.gif"></td>
</tr></table>
<p>

<h2>Kepler's second law</h2>
<table cols=2 width=100% cellpadding=5><tr>
<td><b>The line between the Sun and a planet sweeps over equal areas in equal time.</b></td>
<td><img src="kepler2.gif"></td>
</tr></table>
<p>

<h2>Kepler's third law</h2>
<table cols=2 width=100% cellpadding=5><tr>
<td><b>The cube of the semimajor axis of a planet's orbit is proportional to the square of its period of revolution.</b></td>
<td><img src="kepler3.gif"></td>
</tr></table>
<p>

Kepler's laws are purely phenomenological: They provide a concise description of the observed data, but don't attempt to explain the physical causes of the motion.
<p>
Such a physical explanation was achieved by Sir Isaac Newton, who formulated the three fundamental dynamic laws of motion and the law of gravitation, from which Kepler's laws can be derived.
<p>
Using Newtonian dynamics, it also follows that Kepler's laws are only approximations of the true motions. Strictly they apply only to two-body problems, while in a multi-body system such as the solar system, the influence of other celestial bodies causes perturbations in the orbits. However, in most cases these perturbations are small, so that Kepler's laws provide a good model of reality.
<p>
The derivation from Newton's laws also shows that orbital trajectories are not limited to ellipses, but may also have a parabolic or hyperbolic shape. In general, the orbits in a 2-body problem are described by conic sections. Ellipses describe periodic orbits, while parabolic and hyperbolic orbits are non-periodic.

</BODY>
</HTML>
